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Curvature invariant

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They have limitations because many distinct spacetimes cannot be distinguished on this basis. In particular, so called
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Stephani, Hans (2009). "9. Invariants and the characterization of geometries".
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Two of the most basic curvature invariants in general relativity are the
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using any number of polynomial curvature invariants (of any order).
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An important unsolved problem in general relativity is to give a
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These are analogous to two familiar quadratic invariants of the
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and tensors formed from these by the operations of taking
533: 476: 305: 239: 134:quantities constructed from tensors that represent 46:. Unsourced material may be challenged and removed. 359: 281: 360:{\displaystyle R_{abcd}\,{{}^{\star }\!R}^{abcd}} 337: 553: 496: 435:Exact Solutions of Einstein's Field Equations 8: 560: 546: 503: 489: 342: 331: 329: 327: 325: 310: 304: 264: 259: 244: 238: 169:The invariants most often considered are 106:Learn how and when to remove this message 416:Curvature invariant (general relativity) 192:. However, it can also be considered a 7: 514: 512: 457: 455: 208:) are polynomial scalar invariants. 44:adding citations to reliable sources 282:{\displaystyle R_{abcd}\,R^{abcd}} 14: 516: 459: 204:, whose coefficients and roots ( 138:. These tensors are usually the 20: 411:Carminati–McLenaghan invariants 374:in classical electromagnetism. 31:needs additional citations for 218:metric theories of gravitation 1: 294:Chern–Pontryagin scalar 165:Types of curvature invariants 532:. You can help Knowledge by 475:. You can help Knowledge by 372:electromagnetic field tensor 605: 511: 454: 159:covariant differentiations 124:pseudo-Riemannian geometry 589:Riemannian geometry stubs 406:Cartan–Karlhede algorithm 202:characteristic polynomial 188:of fourth rank acting on 184:The Riemann tensor is a 200:, and as such it has a 179:differential invariants 528:-related article is a 471:-related article is a 361: 283: 362: 284: 212:Physical applications 171:polynomial invariants 55:"Curvature invariant" 303: 237: 186:multilinear operator 175:quadratic invariants 128:curvature invariants 40:improve this article 579:Riemannian geometry 526:Riemannian geometry 421:Ricci decomposition 394:Minkowski spacetime 120:Riemannian geometry 357: 279: 229:Kretschmann scalar 222:general relativity 541: 540: 484: 483: 116: 115: 108: 90: 596: 584:Relativity stubs 562: 555: 548: 520: 513: 505: 498: 491: 463: 456: 448: 366: 364: 363: 358: 356: 355: 341: 336: 335: 330: 324: 323: 288: 286: 285: 280: 278: 277: 258: 257: 111: 104: 100: 97: 91: 89: 48: 24: 16: 604: 603: 599: 598: 597: 595: 594: 593: 569: 568: 567: 566: 510: 509: 452: 445: 432: 429: 402: 328: 326: 306: 301: 300: 260: 240: 235: 234: 214: 194:linear operator 190:tangent vectors 167: 112: 101: 95: 92: 49: 47: 37: 25: 12: 11: 5: 602: 600: 592: 591: 586: 581: 571: 570: 565: 564: 557: 550: 542: 539: 538: 521: 508: 507: 500: 493: 485: 482: 481: 464: 450: 449: 444:978-0521467025 443: 428: 425: 424: 423: 418: 413: 408: 401: 398: 390:VSI spacetimes 368: 367: 354: 351: 348: 345: 340: 334: 322: 319: 316: 313: 309: 290: 289: 276: 273: 270: 267: 263: 256: 253: 250: 247: 243: 213: 210: 166: 163: 140:Riemann tensor 114: 113: 28: 26: 19: 13: 10: 9: 6: 4: 3: 2: 601: 590: 587: 585: 582: 580: 577: 576: 574: 563: 558: 556: 551: 549: 544: 543: 537: 535: 531: 527: 522: 519: 515: 506: 501: 499: 494: 492: 487: 486: 480: 478: 474: 470: 465: 462: 458: 453: 446: 440: 436: 431: 430: 426: 422: 419: 417: 414: 412: 409: 407: 404: 403: 399: 397: 395: 391: 386: 384: 380: 375: 373: 352: 349: 346: 343: 338: 332: 320: 317: 314: 311: 307: 299: 298: 297: 295: 274: 271: 268: 265: 261: 254: 251: 248: 245: 241: 233: 232: 231: 230: 225: 223: 219: 211: 209: 207: 203: 199: 195: 191: 187: 182: 180: 176: 172: 164: 162: 160: 156: 153: 149: 145: 141: 137: 133: 129: 125: 121: 110: 107: 99: 88: 85: 81: 78: 74: 71: 67: 64: 60: 57: â€“  56: 52: 51:Find sources: 45: 41: 35: 34: 29:This article 27: 23: 18: 17: 534:expanding it 523: 477:expanding it 466: 451: 434: 387: 376: 369: 293: 291: 226: 215: 183: 178: 174: 170: 168: 155:contractions 148:Ricci tensor 127: 117: 102: 93: 83: 76: 69: 62: 50: 38:Please help 33:verification 30: 206:eigenvalues 144:Weyl tensor 96:August 2009 573:Categories 469:relativity 427:References 196:acting on 66:newspapers 381:(and any 333:⋆ 198:bivectors 136:curvature 400:See also 383:syzygies 292:and the 220:such as 80:scholar 441:  146:, the 142:, the 132:scalar 82:  75:  68:  61:  53:  524:This 467:This 379:basis 87:JSTOR 73:books 530:stub 473:stub 439:ISBN 157:and 152:dual 130:are 122:and 59:news 216:In 118:In 42:by 575:: 296:, 181:. 161:. 126:, 561:e 554:t 547:v 536:. 504:e 497:t 490:v 479:. 447:. 353:d 350:c 347:b 344:a 339:R 321:d 318:c 315:b 312:a 308:R 275:d 272:c 269:b 266:a 262:R 255:d 252:c 249:b 246:a 242:R 109:) 103:( 98:) 94:( 84:· 77:· 70:· 63:· 36:.

Index


verification
improve this article
adding citations to reliable sources
"Curvature invariant"
news
newspapers
books
scholar
JSTOR
Learn how and when to remove this message
Riemannian geometry
pseudo-Riemannian geometry
scalar
curvature
Riemann tensor
Weyl tensor
Ricci tensor
dual
contractions
covariant differentiations
multilinear operator
tangent vectors
linear operator
bivectors
characteristic polynomial
eigenvalues
metric theories of gravitation
general relativity
Kretschmann scalar

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